(a) Explain how a vector field

$$V=\xi(x, u) \frac{\partial}{\partial x}+\eta(x, u) \frac{\partial}{\partial u}$$

generates a 1-parameter group of transformations $g^{\epsilon}:(x, u) \mapsto(\tilde{x}, \tilde{u})$ in terms of the solution to an appropriate differential equation. [You may assume the solution to the relevant equation exists and is unique.]

(b) Suppose now that $u=u(x)$. Define what is meant by a Lie-point symmetry of the ordinary differential equation

$$\Delta\left[x, u, u^{(1)}, \ldots, u^{(n)}\right]=0, \quad \text { where } \quad u^{(k)} \equiv \frac{d^{k} u}{d x^{k}}, \quad k=1, \ldots, n$$

(c) Prove that every homogeneous, linear ordinary differential equation for $u=u(x)$ admits a Lie-point symmetry generated by the vector field

$$V=u \frac{\partial}{\partial u}$$

By introducing new coordinates

$$s=s(x, u), \quad t=t(x, u)$$

which satisfy $V(s)=1$ and $V(t)=0$, show that every differential equation of the form

$$\frac{d^{2} u}{d x^{2}}+p(x) \frac{d u}{d x}+q(x) u=0$$

can be reduced to a first-order differential equation for an appropriate function.